Mathematical Principles of Theoretical Physics

1.5 Unified Field Theory Based On PID and PRI

with the principle of general relativity. Also, there are nobasic principles and rules for com-
bining relativistic systems and the Galilean systems together to form a consistent system. The
distinction between relativistic and Galilean systems gives rise to an obstacle for establishing
a consistent model of astrophysical dynamics. This difficulty can be circumvented by using
the above mentioned symmetry-breaking principle, where wehave to chose the coordinate
system
xμ= (x^0 ,x), x^0 =ctandx= (x^1 ,x^2 ,x^3 ),

such that the metric is in the form:

ds^2 =−

(

1 +

2

c^2

ψ(x,t)

)

c^2 dt^2 +gij(x,t)dxidxj.

Heregij( 1 ≤i,j≤ 3 )are the spatial metric, andψrepresents the gravitational potential.
The resulting system breaks the symmetry of general coordinate transformations, and we call
such symmetry-breaking as relativistic-symmetry breaking.

1.5 Unified Field Theory Based On PID and PRI

One of the greatest problems in physics is to unify all four fundamental interactions. Albert
Einstein was the first person who made serious attempts to this problem.
Most attempts so far have focused on unification through large symmetry, following Ein-
stein’s vision. However, as indicated above, one of the mostprofound implication of PRI is
that such a unification with a large symmetry group would violate PRI, which is a basic logic
requirement. In fact, the unification should be based on coupling different interactions using
the principle of symmetry-breaking (PSB) instead of seeking for a large symmetry group.
The basic principles for the four fundamental interactionsaddressed in the previous sec-
tions have demonstrated that the three first principles, PID, PRI and PSB, offer an entirely
different route for the unification, which is one of the main aims of this book:

1) the general relativity and the gauge symmetries dictate theLagrangian;

2) the coupling of the four interactions is achieved through PID, PRI and

PSB in the unified field equations, which obey the PGR and PRI, but break

spontaneously the gauge symmetry; and

3) the unified field model can be easily decoupled to study individual interac-

tion, when the other interactions are negligible.

Hereafter we address briefly the main ingredients of the unified field theory.

Lagrangian action

Following the simplicity principle of laws of Nature as stated in Principle2.2, the three ba-
sic symmetries—the Einstein general relativity, the Lorentz invariance and the gauge invariance—
uniquely determine the interaction fields and their Lagrangian actions for the four interac-
tions:

• The Lagrangian action for gravity is the Einstein-Hilbert functional given by (1.2.2);